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Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so. For instance, to show that ‘{1, padma, blue} is a set’ is true, you need to say that this is true because it is a well-defined collection of 3 objects.) i) For any two sets A and B, A B c A ∩ = B . ii) The matrix 0 0 1 1 is singular. iii) The contrapositive of ‘ ∃ y such that )y(P is true’ is ‘ ∈Z ∃ x such that )x(P ∈Z is true’. iv) The system 1 2x − 3y = and 0 6y − 4x + 2 = has a unique solution. v) If y,x such that y ∈C x 2 = and x y 2 = , then 1 x = y = . vi) a ≥ b ⇔ −a ≤ −b is an absolute inequality. vii) If }2 A = ϕ,B = },2,1{ C = {− ,1 − , then A× B×C has 4 elements. viii) The argument of 1+ i3 is 3 π . ix) A linear equation over R can have at most one root in C R . x) x1 − x2 = x1 − x2 ∀x1 x, 2 ∈R .
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